The simplicial boundary of a CAT(0) cube complex

Hagen, Mark F

doi:10.2140/agt.2013.13.1299

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The simplicial boundary of a CAT(0) cube complex

Mark F Hagen

Algebraic & Geometric Topology 13 (2013) 1299–1367

DOI: 10.2140/agt.2013.13.1299

Abstract

For a CAT(0) cube complex $X$ , we define a simplicial flag complex $\partial_{△} X$ , called the simplicial boundary, which is a natural setting for studying nonhyperbolic behavior of $X$ . We compare $\partial_{△} X$ to the Roller, visual and Tits boundaries of $X$ , give conditions under which the natural CAT(1) metric on $\partial_{△} X$ makes it isometric to the Tits boundary, and prove a more general statement relating the simplicial and Tits boundaries. The simplicial boundary $\partial_{△} X$ allows us to interpolate between studying geodesic rays in $X$ and the geometry of its contact graph $Γ X$ , which is known to be quasi-isometric to a tree, and we characterize essential cube complexes for which the contact graph is bounded. Using related techniques, we study divergence of combinatorial geodesics in $X$ using $\partial_{△} X$ . Finally, we rephrase the rank-rigidity theorem of Caprace and Sageev in terms of group actions on $Γ X$ and $\partial_{△} X$ and state characterizations of cubulated groups with linear divergence in terms of $Γ X$ and $\partial_{△} X$ .

Keywords

CAT(0) cube complex, contact graph, divergence, rank-one isometry, simplicial boundary

Mathematical Subject Classification 2010

Primary: 05C25, 20F65, 57M99

References

Publication

Received: 4 January 2012

Revised: 1 May 2012

Accepted: 20 December 2012

Published: 24 April 2013

Authors

Mark F Hagen
Department of Mathematics University of Michigan 530 Church Street 1859 East Hall Ann Arbor, MI 48109 USA
http://www-personal.umich.edu/~mfhagen/index.html

Volume 13, issue 3 (2013)